You could send a short radio pulse toward a black hole and wait for the reflected echo. Intuitively, the shorter the signal, the better it would probe the finest details of spacetime right at the horizon: that region from which even light cannot return. But nature sets a trap: an ultra-short wave almost flies straight through the gravitational barrier and disappears into the hole. Only a faint whisper comes back.

In this new work, the paradox is treated as a balancing act. A pulse that is too long reflects well but smears the picture, like a blurry photograph. One that is too short gives sharpness but barely returns. The authors found the sweet spot: the pulse width should be roughly the inverse square root of the height of the black hole's potential barrier. Such a signal extracts the maximum information about metric distortions near the horizon from the reflected echo.

To test the idea, they used a simple Pöschl–Teller potential model, where everything can be calculated analytically. It turned out that the black hole's fundamental oscillations (quasinormal modes) carry only part of the information, while the full picture comes from the entire reflected signal — including the first, fastest response. This creates an unexpected bridge between black hole spectroscopy, semiclassical scattering, and information theory.